Exploring the solutions of a financial bubble model via a new fractional derivative

Sabri T. M. Thabet; Reem M. Alraimy; Imed Kedim; Aiman Mukheimer; Thabet Abdeljawad; Sabri T. M. Thabet; Reem M. Alraimy; Imed Kedim; Aiman Mukheimer; Thabet Abdeljawad

doi:10.3934/math.2025394

AIMS Mathematics

2025, Volume 10, Issue 4: 8587-8614. doi: 10.3934/math.2025394

Previous Article Next Article

Research article Special Issues

Exploring the solutions of a financial bubble model via a new fractional derivative

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
2.
Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen
3.
Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02814, Korea
4.
Department of Mathematics, Education College-Aden, University of Aden, Aden, Yemen
5.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
6.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
7.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
8.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea
9.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa
10.
Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally 32093, Kuwait

Received: 03 January 2025 Revised: 15 March 2025 Accepted: 02 April 2025 Published: 14 April 2025
MSC : 26A33, 91G80, 93D05

This study presents a fractional mathematical model that explains how behavioral and social contagion in the market, can explain the bubble, collapse, and stability phases of financial bubbles. We study the proposed model via a new fractional derivative in the framework of the Caputo derivative involving a modified generalized Mittag-Leffler function (MLF). Furthermore, we use the Schauder and Banach fixed point theorems (FPTs) to prove the existence and uniqueness (E & U) of the solution of the model. Moreover, we discover the equilibrium point and identify the nullcline points of the suggested model. Then we use the Lyapunov function to investigate the global stability of the discovered equilibrium point at certain criteria, leading to the discovery of a globally stable solution. To obtain numerical results, we use the fractional Adams-Bashforth technique of order 3. We also analyze the residual error to evaluate the correctness of the proposed method. After that, we perform simulations with different parameter values and fractional orders to show the applicability of the method in different contexts. Additionally, our results can be applied to the fractional generalized Atangana-Baleanu-Caputo (GABC), Atangana-Baleanu-Caputo (ABC), and Caputo–Fabrizo-Caputo (CFC) derivatives as special cases at certain parameters. The results confirm that the technique can produce accurate answers in many settings.
- financial bubbles,
- fractional derivatives,
- existence and uniqueness,
- Lyapunov equation,
- Adams-Bashforth method
Citation: Sabri T. M. Thabet, Reem M. Alraimy, Imed Kedim, Aiman Mukheimer, Thabet Abdeljawad. Exploring the solutions of a financial bubble model via a new fractional derivative[J]. AIMS Mathematics, 2025, 10(4): 8587-8614. doi: 10.3934/math.2025394

Related Papers:

Abstract

This study presents a fractional mathematical model that explains how behavioral and social contagion in the market, can explain the bubble, collapse, and stability phases of financial bubbles. We study the proposed model via a new fractional derivative in the framework of the Caputo derivative involving a modified generalized Mittag-Leffler function (MLF). Furthermore, we use the Schauder and Banach fixed point theorems (FPTs) to prove the existence and uniqueness (E & U) of the solution of the model. Moreover, we discover the equilibrium point and identify the nullcline points of the suggested model. Then we use the Lyapunov function to investigate the global stability of the discovered equilibrium point at certain criteria, leading to the discovery of a globally stable solution. To obtain numerical results, we use the fractional Adams-Bashforth technique of order 3. We also analyze the residual error to evaluate the correctness of the proposed method. After that, we perform simulations with different parameter values and fractional orders to show the applicability of the method in different contexts. Additionally, our results can be applied to the fractional generalized Atangana-Baleanu-Caputo (GABC), Atangana-Baleanu-Caputo (ABC), and Caputo–Fabrizo-Caputo (CFC) derivatives as special cases at certain parameters. The results confirm that the technique can produce accurate answers in many settings.

References

[1]	G. C. Evans, The dynamics of monopoly, Am. Math. Mon., 31 (1924), 77–83. https://doi.org/10.1080/00029890.1924.11986301 doi: 10.1080/00029890.1924.11986301
[2]	G. Barlevy, Economic theory and asset bubbles, Econ. Perspect., 31 (2007), 44–59.
[3]	B. Bernanke, M. Gertler, Agency costs, net worth, and business fluctuations, Am. Econ. Rev., 79 (1989), 14–31.
[4]	B. Bernanke, M. Gertler, S. Gilchrist, The financial accelerator in a quantitative business cycle framework, Handb. Macroecon., 1 (1999), 1341–1393. https://doi.org/10.1016/S1574-0048(99)10034-X doi: 10.1016/S1574-0048(99)10034-X
[5]	C. T. Carlstrom, T. S. Fuerst, Agency costs, net worth, and business fluctuations: A computable general equilibrium analysis, Am. Econ. Rev., 87 (1997), 893–910.
[6]	N. Kiyotaki, J. Moore, Credit cycles, J. Political Econ., 105 (1997), 211–248. https://doi.org/10.1086/262072 doi: 10.1086/262072
[7]	Q. He, P. Xia, C. Hu, B. Li, Public information, actual intervention and inflation expectations, Transform. Bus. Econ., 21 (2022), 644–666.
[8]	Q. He, X. Zhang, P. Xia, C. Zhao, S. Li, A comparison research on dynamic characteristics of Chinese and American energy prices, J. Glob. Inf. Manag., 31 (2023), 1–16. http://dx.doi.org/10.4018/JGIM.319042 doi: 10.4018/JGIM.319042
[9]	J. H. He, Seeing with a single scale is always unbelieving: From magic to tow-scale fractal, Therm. Sci., 25 (2021), 1217–1219. https://doi.org/10.2298/TSCI2102217H doi: 10.2298/TSCI2102217H
[10]	J. H. He, Thermal science for the real world: Reality and challenge, Therm. Sci., 24 (2020), 2289–2294. https://doi.org/10.2298/TSCI191001177H doi: 10.2298/TSCI191001177H
[11]	D. Sornette, P. Cauwels, Financial bubbles: Mechanisms and diagnostics, Rev. Behav. Econ., 2 (2015), 279–305. http://dx.doi.org/10.1561/105.00000035 doi: 10.1561/105.00000035
[12]	T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 130. https://doi.org/10.1186/s13660-017-1400-5 doi: 10.1186/s13660-017-1400-5
[13]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. http://dx.doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[14]	T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2018 (2018), 468. https://doi.org/10.1186/s13662-018-1914-2 doi: 10.1186/s13662-018-1914-2
[15]	S. T. M. Thabet, T. Abdeljawad, I. Kedim, M. I. Ayari, A new weighted fractional operator with respect to another function via a new modified generalized Mittag-Leffler law, Bound. Value Probl., 2023 (2023), 100. https://doi.org/10.1186/s13661-023-01790-7 doi: 10.1186/s13661-023-01790-7
[16]	J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[17]	S. T. M. Thabet, M. Vivas-Cortez, I. Kedim, Analytical study of $ABC$-fractional pantograph implicit differential equation with respect to another function, AIMS Mathematics, 8 (2023), 23635–23654. https://doi.org/10.3934/math.20231202 doi: 10.3934/math.20231202
[18]	K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[19]	I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386.
[20]	H. Boulares, A. Moumen, K. Fernane, J. Alzabut, H. Saber, T. Alraqad, et al., On solutions of fractional integrodifferential systems involving ${\rm{ \mathsf{ ψ}}}$-Caputo derivative and ${\rm{ \mathsf{ ψ}}}$-Riemann-Liouville fractional integral, Mathematics, 11 (2023), 1465. https://doi.org/10.3390/math11061465 doi: 10.3390/math11061465
[21]	G. Mani, A. J. Gnanaprakasam, O. Ege, A. Aloqaily, N. Mlaiki, Fixed point results in $c^$-algebra-valued partial $b$-metric spaces with related application, Mathematics*, 11 (2023), 1158. https://doi.org/10.3390/math11051158 doi: 10.3390/math11051158
[22]	G. Mani, A. J. Gnanaprakasam, L. Guran, R. George, Z. D. Mitrović, Some results in fuzzy $b$-metric space with $b$-triangular property and applications to fredholm integral equations and dynamic programming, Mathematics, 11 (2023), 4101. https://doi.org/10.3390/math11194101 doi: 10.3390/math11194101
[23]	G. Mani, S. Haque, A. J. Gnanaprakasam, O. Ege, N. Mlaiki, The study of bicomplex-valued controlled metric spaces with applications to fractional differential equations, Mathematics, 11 (2023), 2742. https://doi.org/10.3390/math11122742 doi: 10.3390/math11122742
[24]	X. Wang, J. Alzabut, M. Khuddush, M. Fečkan, Solvability of iterative classes of nonlinear elliptic equations on an exterior domain, Axioms, 12 (2023), 474. https://doi.org/10.3390/axioms12050474 doi: 10.3390/axioms12050474
[25]	A. G. M. Selvam, D. Baleanu, J. Alzabut, D. Vignesh, S. Abbas, On Hyers-Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum, Adv. Differ. Equ., 2020 (2020), 456. https://doi.org/10.1186/s13662-020-02920-6 doi: 10.1186/s13662-020-02920-6
[26]	A. Boutiara, S. Etemad, S. T. M. Thabet, S. K. Ntouyas, S. Rezapour, J. Tariboon, A mathematical theoretical study of a coupled fully hybrid $(\kappa, \phi)$-fractional order system of BVPs in generalized banach spaces, Symmetry, 15 (2023), 1041. https://doi.org/10.3390/sym15051041 doi: 10.3390/sym15051041
[27]	S. T. M. Thabet, S. Al-Sa'di, I. Kedim, A. S. Rafeeq, S. Rezapour, Analysis study on multi-order $\varrho$-Hilfer fractional pantograph implicit differential equation on unbounded domains, AIMS Mathematics, 8 (2023), 18455–18473. https://doi.org/10.3934/math.2023938 doi: 10.3934/math.2023938
[28]	K. Diethelm, The analysis of fractional differential equations, Berlin, Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[29]	C. Li, S. Sarwar, Existence and continuation of solution for Caputo type fractional differential equations, Electron. J. Differ. Equ., 2016 (2016), 1–14.
[30]	V. Lakshmikantham, S. Leela, M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal., 12 (2008), 365–376.
[31]	V. Lakshmikantham, S. Leela, J. Vasundara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
[32]	T. A. Burton, Fractional differential equations and Lyapunov functionals, Nonlinear Anal. Theory Methods Appl., 74 (2011), 5648–5662. https://doi.org/10.1016/j.na.2011.05.050 doi: 10.1016/j.na.2011.05.050
[33]	K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algor., 36 (2004), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be doi: 10.1023/B:NUMA.0000027736.85078.be
[34]	A. Afilipoaei, G. Carrero, A mathematical model of financial bubbles: A behavioral approach, Mathematics, 11 (2023), 4102. http://dx.doi.org/10.3390/math11194102 doi: 10.3390/math11194102
[35]	T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals, Chaos, 29 (2019), 023102. https://doi.org/10.1063/1.5085726 doi: 10.1063/1.5085726
[36]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[37]	R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229 (2009), 392–399. https://doi.org/10.1016/j.cam.2008.04.004 doi: 10.1016/j.cam.2008.04.004
[38]	A. Granas, J. Dugundji, Fixed point theory, New York, Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
[39]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014. https://doi.org/10.1142/9069
[40]	S. Deepika, P. Veeresha, Dynamics of chaotic waterwheel model with the asymmetric flow within the frame of Caputo fractional operator, Chaos Soliton Fract., 169 (2023), 113298. https://doi.org/10.1016/j.chaos.2023.113298 doi: 10.1016/j.chaos.2023.113298
[41]	A. Atangana, S. I. Araz, A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton polynomial, Alex. Eng. J., 60 (2021), 3781–3806. https://doi.org/10.1016/j.aej.2021.02.016 doi: 10.1016/j.aej.2021.02.016
[42]	R. Douaifia, S. Bendoukha, S. Abdelmalek, A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study, Math. Comput. Simul., 187 (2021), 391–413. https://doi.org/10.1016/j.matcom.2021.03.009 doi: 10.1016/j.matcom.2021.03.009
[43]	F. Khan, M. Omar, G. Mustafa, Generalized family of approximating schemes based on Newton interpolating polynomials and its error analysis, J. Math., 51 (2019), 81–95.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)